656 Acta Cryst. (1987). A43, 656-674 Multiplicity Distribution of Reflections in Laue Diffraction BY D. W. J. CRUICKSHANK,* J. R. HELLIWELL AND K. MOFFATt Department of Physics, University of York, Heslington, York YO 1 5DD, England and SERC, Daresbury Laboratory, Daresbury, Warrington, Cheshire WA4 4AD, England (Received 6 January 1987; accepted 2 April 1987) Abstract If a crystal is illuminated by a polychromatic beam of X-rays, then many orders of each Bragg reflection may be stimulated simultaneously, and overlap exactly in scattering angle. The overlap of these multiple orders along a ray (a central line in reciprocal space) poses a problem for Laue methods. A theory of the distribution of multiple orders as a function of the relevant experimental parameters is presented, with the following conclusions: (1) If the angular acceptance of the detector is unrestricted, then a remarkably large proportion (72.8%) of all Bragg reflections occur on single rays for the case of an infinite range of incident wavelengths. (2) This pro- portion increases to greater than 83% when more realistic experimental values of hmax and '~min are used. (3) This proportion depends only on the ratio of ~max to '~min and not on the space group, unit-cell dimensions, crystal orientation or the limiting resolu- tion of the crystal d'max (provided d*ax < 2/hmax). (4) The total number of single rays, like the total number of all stimulated Bragg reflections, is approximately proportional to the wavelength range. (5) The propor- tion of reflections at a given resolution d* that lie on single or double rays depends markedly on d*, and on the ratio of Amax to Ami,; it is generally lower at low resolution than at high. (6) Restricted angular acceptance of the detector can reduce significantly both the proportion and the total number of single rays. (7) Agreement between the theoretical distribu- tions and those derived from analysis of X-ray Laue photographs of macromolecular crystals, and from extensive computer simulations, is good. It is evident that, under a wide variety of experimental conditions, the effect of multiple orders is not a serious limitation on the use of the Laue method for structure determi- nation. The analysis presented has some relevance to polychromatic neutron diffraction. * Also Department of Chemistry, UMIST, Manchester, M60 1QD, England. t Also Section of Biochemistry, Molecular and Cell Biology, Cornell University, Ithaca, New York 14853, USA. 0108-7673/87/050656-19501.50 I. Introduction The recent availability of synchrotron X-ray sources has renewed interest in Laue diffraction methods, which exploit directly the polychromatic nature of such sources. In preliminary studies, Laue diffraction from protein crystals (Moffat, Szebenyi & Bilderback, 1984; Moffat, Schildkamp, Bilderback & Volz, 1986; Bilderback, Moffat & Szebenyi, 1984; Helliwell, 1984, p. 1468, 1985; Hedman, Hodgson, Helliwell, Lidding- ton & Papiz, 19~'5) and from small inorganic crystals (Wood, Thompson & Matthewman, 1983; Hails, Harding, Helliwell, Liddington & Papiz, 1984; Harding et al., in preparation) has been examined. These studies suggest that the Laue method possesses advantages over more conventional monochromatic data collection methods. It makes optimum use of the synchrotron radiation spectrum, and affords a reduction in exposure time of several orders of magni- tude. The Laue method thus permits very brief exposures in the millisecond time range on strongly scattering protein samples (Bilderback et al., 1984; Hajdu, Machin, Campbell, Clifton, Zurek, Gover & Johnson, 1986; Moffat et al., 1986) and the examination of microcrystals (Hedman et al., 1985). A stationary crystal yields integrated intensities directly, which are relatively insensitive to transient changes in unit-cell dimensions or crystal orientation. A typical Laue diffraction pattern contains many more reflections than a typical monochromatic oscillation pattern. These advantages are particularly appropriate for dynamic experiments, in which the diffraction intensities change rapidly with time in response to a structural perturbation (Wood et al., 1983; Moffat et al., 1984, 1986; Helliwell, 1985). A fundamental complexity of the Laue method is the multiple-orders problem, which is revealed when Bragg's law is applied to the diffraction of polychro- matic X-rays. If a crystal contains a spacing d, it also contains spacings d/2, d/3,.., or, in general, d/j, where j is any positive integer. Then Bragg's law is simultaneously satisfied by the set of values (d, A), (d/2, A/2), (d/3, A/3) ... (d/j, A/j) .... That is, all orders of a Bragg reflection are exactly superimposed (apart from very small dispersive effects). The reciprocal-lattice points corresponding to the first and all higher orders lie on a central line passing O 1987 International Union of Crystallography